1/3/2023

Definition

In mathematics, the Fibonacci sequence is made up from Fibonacci numbers, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are here:

\[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.\]

A bit of history

The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci (Sigler 2002) where it is used to calculate the growth of rabbit populations (Hemenway 2005). Fibonacci considers the growth of an idealized rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever.

Publications

The Fibbonacci sequence is popular topic of research and below you can see a table with a few examples of such publicationas.

Fibbonacci sequence in equations

\[\begin{equation} \begin{aligned} \text{Given} \,\, f_1=1,f_2=1, \text{then}\\ f_3=f_2+f_1=1+1=2,\\ f_4=f_3+f_2=2+1=3,\\ f_5=f_4+f_3=3+2=5, \end{aligned} \end{equation}\]

Interactive plot

Slide with R

# take input from the user
nterms = 10
# first two terms
n1 = 0
n2 = 1
count = 2
# check if the number of terms is valid
if(nterms <= 0) {
print("Plese enter a positive integer")
} else {
if(nterms == 1) {
print("Fibonacci sequence:")
print(n1)
} else {
print("Fibonacci sequence:")
print(n1)
print(n2)
while(count < nterms) {
nth = n1 + n2
print(nth)
# update values
n1 = n2
n2 = nth
count = count + 1
}
}
}

Slide with Plot

Here we can see an animation of spirals generated with Fibonacci sequences.

Fibonacci spirals

Fibonacci spirals

References

Hemenway, Priya. 2005. Divine Proportion: Phi in Art, Nature, and Science. New York: Published by Sterling Pub. Co.

Sigler, Laurence. 2002. Fibonacci’s Liber Abaci. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4613-0079-3.